Frequency response masking (FRM) is an efficient method for designing an FIR (Finite Impulse Response) filter having a narrow transition band characteristic. When a filter has a narrow transition band, multi-stage FRM may be utilized to further reduce its complexity.
A filter is composed of a prototype filter Ha(z) and two masking filters Hma(z), Hmc(z). The transition band of the filter is provided by an interpolation filter Ha(zM) or its complement
  (            z              -                              M            ⁡                          (                                                N                  a                                -                1                            )                                2                      -                  H        a            ⁡              (                  z          M                )              )where M is an interpolation factor for Ha(z). The purpose of using two masking filters Hma(z) and Hmc(z) is to remove unnecessary periodic sub-bands.
Two-stage FRM filters are widely used in practice. The structure of a two-stage FRM filter is as shown in FIG. 2. A transition band shaping filter is represented by G(z). A constraint condition for interpolation factors M, P, Q is as follows:M=kP=kQ  (3).
At present, various improvements made to the two-stage FRM filters are based on the assumption of the equation (3). The satisfaction of the equation (3) leads to a direct problem that the second-stage output must be a periodic amplitude response. However, the second-stage output does not have to be periodic as long as it provides a desired transition band for a target filter. Therefore, the complexity of an FRM filter can be further reduced if the values of the three factors can be found in a wider range.